The Minimal Polynomial (Algebra 3: Lecture 1 Video 3)

Lecture 1: We started this lecture by recalling the Classification of Modules over a PID, Existence- Invariant Factor Form (Theorem 5 from Section 12.1). We also recalled some material about F[x]-modules from Section 10.1. We know that if V is a finite dimensional vector space and T is a linear transformation on V, then the pair (V,T) defines an F[x]-module. Applying the classification theorem shows that this is a torsion F[x]-module, so can be described in terms of invariant factors, a_1(x),...,a_m(x), which we can take to be monic polynomials in F[x]. We gave several definitions related to eigenvalues, eigenvectors, and eigenspaces. This led to a definition of the characteristic polynomial of a linear transformation and the characteristic polynomial of an n x n matrix. We saw that similar matrices have the same characteristic polynomial. We defined the minimal polynomial of a linear transformation and the minimal polynomial of an n x n matrix. We saw that apply the classification theorem for modules over a PID showed that the minimal polynomial of an F[x]-module V is its largest invariant factor. We proved that similar matrices have the same minimal polynomial. We saw how a nice choice of basis for each cyclic F[x]-module F[x]/(a(x)) led to a simple matrix representation for the 'multiplication by x' linear transformation. This led to the definition of the companion matrix of a polynomial. We saw how to choose a nice basis for V so that the matrix of the linear transformation T with respect to this basis was a direct sum of the companion matrices of the invariant factors of V. This led to the definition of the Rational Canonical Form of a linear transformation and the Rational Canonical Form of a matrix.

Reading: In this lecture we closely followed the beginning of Section 12.2 of Dummit and Foote (pages 472-475), but we included more direct references to material that we covered in 206B. For example, we recalled results from Sections 12.1, 11.2, and 10.1. We also included some additional results about similar matrices and characteristic and minimal polynomials. For example, we gave a proof of Exercise 1 in Section 12.2.